## Arc Length

Arc length is better defined as the distance along the part of the circumference of any circle or any curve (arc). Any distance along the curved line that makes up the arc is known as the arc length. A part of a curve or a part of a circumference of a circle is called Arc. All of them have a curve in their shape. The length of an arc is longer than any straight line distance between its endpoints (a chord).

## What is Arc Length?

The** **arc length is defined as the interspace between the two points along a section of a curve. An arc of a circle is any part of the circumference. The angle subtended by an arc at any point is the angle formed between the two line segments joining that point to the end-points of the arc. For example, in the circle shown below, OP is the arc of the circle with center Q. The arc length of this arc OP is given as L.

## Arc Length Formula

The length of an arc can be calculated using different formulas, based on the unit of the central angle of the arc. The measurements of the central angle can be given in degrees or radians, and accordingly, we calculate the arc length of a circle. For a circle, the arc length formula is θ times the radius of a circle.

The arc length formula in radians can be expressed as, arc length = θ × r, when θ is in radian. Arc Length = θ × (π/180) × r, where θ is in degree, where,

- L = Length of an Arc
- θ = Central angle of Arc
- r = Radius of the circle

### Arc Length Formula in Radians

The arc length of a circle can be calculated using different formulas, based on the unit of the center angle of the arc. The arc length formula in radians can be expressed as,

Arc Length = θ × r

where,

- L = Arc Length
- θ = Center angle of the arc in radians
- r = Radius of the circle

### Denotations in the Arc Length Formula

- s is the arc length
- r is the radius of the circle
- θ is the central angle of the arc

### How to Find Arc Length of a Curve?

The arc length of an arc of a circle can be calculated using different methods and formulas based on the given data. Some important cases are given below,

- find arc length with the radius and central angle
- find arc length without the radius
- find arc length without the central angle

### How to Find Arc Length With the Radius and Central Angle?

The arc length of a circle can be calculated with the radius and central angle using the arc length formula,

- Length of an Arc = θ × r, where θ is in radian.
- Length of an Arc = θ × (π/180) × r, where θ is in degree.

### How to Find Arc Length Without the Radius?

The arc length of a circle can be calculated without the radius using:

**Central angle and the sector area:**

- Multiply the sector area by 2 and further, divide the result by the central angle in radians.
- Find the square root of the result of the division.
- Multiply this obtained root by the central angle again to get the arc length.
- The units of this calculated arc length will be the square root of the sector area units.

Example: Calculate the arc length of a curve with sector area 25 square units and the central angle as 2 radians.

We have,

Sector area = 25 units

Central angle = 2 radians

**Step 1:**Sector area × 2 = 25 × 2 = 50**Step 2:**50/central angle = 50/2 = 25**Step 3:**√25 = 5**Step 4:**5 × central angle = 5 × 2**= 10 units**

Thus, arc length = 10 units

**Central angle and the chord length:**

- Divide the central angle in radians by 2 and further, perform the sine function on it.
- Divide the given chord length by twice the result of step 1. This calculation gives you the radius as result.
- Multiply the radius by the central angle to get the arc length.

**Example**: Calculate the arc length of a curve, whose endpoints touch a chord of the circle measuring 5 units. The central angle subtended by the arc is 2 radians.

We have,

Chord length = 5 units

Central angle = 2 radians

**Step 1:**Central angle/2 = 2/2 = 1**Step 2:**Sin(1) = 0.841**Step 3:**Chord length/ (2 × 0.841) = 5/ 1.682 = 2.973 units = radius**Step 4:**Arc length = radius × central angle = 2.973 × 2 = 5.946 units

Thus, arc length = 5.946 units

### How to Find Arc Length Without the Central Angle?

The arc length of a circle can be calculated without the angle using:

**Radius and the sector area**:

- Multiply the sector area by 2.
- Then divide the result by the radius squared (the units should be the same) to get the central angle in radians.
- Multiply the central angle by the radius to get the arc length.

Example: Calculate the arc length of a curve with sector area 25 square units and radius as 2 units.

We have,

Sector area = 25 units

Central angle = 2 units

**Step 1:**Sector area × 2 = 25 × 2 = 50**Step 2:**50/radius^{2}= 50/4 = 12.5 = central angle(rad)**Step 3:**Arc length = radius × central angle = 2 × 12.5**= 25 units**

Thus, arc length = 25 units

**Radius and chord length:**

- Divide the chord length by twice the given radius.
- Find the inverse sine of the obtained result.
- Double the result of the inverse sine to get the central angle in radians.
- Multiply the central angle by the radius to get the arc length.

**Example**: Calculate the arc length of a curve, whose endpoints touch a chord of the circle measuring 5 units. The radius of the circle is 2 units.

We have,

Chord length = 5 units

Central angle = 2 units

**Step 1:**Chord length/(2 × radius) = 5/(2 × 2) = 1.25**Step 2:**Sin^{-1}(1.25) = 0.949**Step 3:**Central angle = 2 × 0.949 = 1.898 radians**Step 4:**Arc length = radius × central angle = 2 × 1.898**= 3.796 units**

Thus, arc length = 3.796 units

**☛ Important Notes on Arc Length**

Given below are key highlights on the concept of arc length.

- Arc Length = θ × r, where θ is in radian.
- Arc Length = θ × (π/180) × r, where θ is in degree.

### Example Questions Using the Formula for Arc Length

**Question 1:** Calculate the length of an arc if the radius of an arc is 8 cm and the central angle is 40°.

**Solution:**

Radius, r = 8 cm

Central angle, θ = 40°

Arc length = 2 π r × (θ/360°)

So, s = 2 × π × 8 × (40°/360°)

= 5.582 cm

**Question 2: **What is the arc length for a function f(x) = 6 between x = 4 and x = 6?

**Solution:**

Since the function is a constant, the differential of it will be 0. So, the arc length will now be-

So, arc length (s) = (6 – 4) = 2.

### Practice Questions Based on Arc Length Formula

- What would be the length of the arc formed by 75° of a circle having the diameter of 18 cm?
- The length of an arc formed by 60° of a circle of radius “r” is 8.37 cm. Find the radius (r) of that circle.
- Calculate the perimeter of a semicircle of radius 1. cm using the arc length formula.

### Arc Length Calculator

This arc length calculator is a tool that can calculate the length of an arc and the area of a circle sector. This article explains the arc length formula in detail and provides you with step-by-step instructions on how to find the arc length. You will also learn the equation for sector area.

The length of an arc depends on the radius of a circle and the central angle `θ`

. We know that for the angle equal to 360 degrees (2π), the arc length is equal to circumference. Hence, as the proportion between angle and arc length is constant, we can say that:`L / θ = C / 2π`

As circumference `C = 2πr`

,`L / θ = 2πr / 2π`

`L / θ = r`

We find out the arc length formula when multiplying this equation by θ:`L = r * θ`

Hence, the arc length is equal to radius multiplied by the central angle (in radians).

### Area of a sector of a circle

We can find the area of a sector of a circle in a similar manner. We know that the area of the whole circle is equal to πr². From the proportions,`A / θ = πr² / 2π`

`A / θ = r² / 2`

The formula for the area of a sector is:`A = r² * θ / 2`

### How to find the length of an arc and sector area: an example

- Decide on the radius of your circle. For example, it can be equal to 15 cm. (You can also input the diameter into the arc length calculator instead.)
- What will be the angle between the ends of the arc? Let’s say it is equal to 45 degrees, or π/4.
- Calculate the arc length according to the formula above:
`L = r * θ = 15 * π/4 = 11.78 cm`

. - Calculate the area of a sector:
`A = r² * θ / 2 = 15² * π/4 / 2 = 88.36 cm²`

. - You can also use the arc length calculator to find the central angle or the circle’s radius. Simply input any two values into the appropriate boxes and watch it conducting all calculations for you.

### Examples on Arc length

**Example 1: Find the length of an arc cut off by a central angle of 4 radians in a circle with a radius of 6 inches.**

**Solution:**

Center angle, θ = 4 radians, radius, r = 6 inches . Use the arc length formula, L = θ × r = 4 × 6 = 24 inches. ∴ Arc length (PQ) = 24 inches

**Question 2: The radius of the circle is 14 units and the arc subtends 65° at the center. What is the length of the arc?**

**Solution: **We know that,

Circumference of circle = 2πr

C = 2π × 14 = 28π

arc length = (θ/360) × C = (65°/360°)28π = 15.882 units

**Example 3: Find the length of an arc cut off by a central angle, θ = 40º in a circle with a radius of 4 inches.**

**Solution:**

Radius, r = 4 inches , θ = 40º. Use the arc length formula, L = π × (r) × (θ/180º) = π × (4) × (40º/180º) = 2.79 inches. ∴ Arc length (P0) = 2.79 inches

## FAQs on Arc Length

### What is Arc Length of a Circle?

The arc length of a circle is defined as the interspace between the two points along a section of a curve. An arc of a circle is any part of the circumference. The angle subtended by an arc at any point is the angle formed between the two line segments joining that point to the end-points of the arc.

### How do you Find the Length of an Arc Without the Radius?

The arc length of a circle can be calculated without the radius using:

**Central angle and the sector area:**

- Multiply the sector area by 2 and further, divide the result by the central angle in radians.
- Find the square root of the result of the division.
- Multiply this obtained root by the central angle again to get the arc length.
- The units of this calculated arc length will be the square root of the sector area units.

**Central angle and the chord length:**

- Divide the central angle in radians by 2 and further, perform the sine function on it.
- Divide the given chord length by twice the result of step 1. This calculation gives you the radius as result.
- Multiply the radius by the central angle to get the arc length.

### What do You Understand By Arc Lenght Equation?

There are two equations associated with arc length. Given below are the two arc length equations.

- Arc Length = θ × r, where θ is in radian.
- Arc Length = θ × (π/180) × r, where θ is in degree

### How do you Find Arc Length Using Radians?

The arc length can be calculated when the central angle is given in radians using the formula: Arc Length = θ × r, when θ is in radian.

- L = Arc Length
- θ = Center angle of the arc
- r = Radius of the circle

### Does Arc Length have to be in Radians?

No, arc length cannot be in radians. It is a measurement of distance, so cannot be in radians. The central angle subtended at the center can be in radians, degrees, or arcsecs accordingly.

### How do you Find the Circumference of Arc Length?

When arc length is given with central angle θ then the circumference is calculated as Arc Length (L)/Circumference = θ/360º.

### What is the Length of Major Arc Using Arc Length Formula?

A major arc in a circle is larger than a semicircle. It is measured as larger than 180°. Using formula ℓ = rθ we can find the length of an arc of a circle, where θ is in radian.

### How do you find arc length without the radius?

To calculate arc length without radius, you **need the central angle and the sector area**:

- Multiply the area by 2 and divide the result by the central angle in radians.
- Find the square root of this division.
- Multiply this root by the central angle again to get the arc length.
- The units will be the square root of the sector area units.

Or **the central angle and the chord length**:

- Divide the central angle in radians by 2 and perform the sine function on it.
- Divide the chord length by double the result of step 1. This calculation gives you the radius.
- Multiply the radius by the central angle to get the arc length.

### How do you find arc length using radians?

**Multiply the central angle**in radians by the circle’s radius.**That’s it!**The result is simply this multiplication.

### How do you calculate arc length without the angle?

To calculate arc length without the angle, you **need the radius and the sector area**:

- Multiply the area by 2.
- Then divide the result by the radius squared (make sure that the units are the same) to get the central angle in radians.

Or you can use **the radius and chord length**:

- Divide the chord length by double the radius.
- Find the inverse sine of the result (in radians).
- Double the result of the inverse sine to get the central angle in radians.
- Once you have the central angle in radians, multiply it by the radius to get the arc length.

### Does arc length have to be in radians?

**Arc length is a measurement of distance, so it cannot be in radians**. The central angle, however, does not have to be in radians. It can be in any unit for angles you like, from degrees to arcsecs. **Using radians**, however, **is much easier for calculations** regarding arc length, as finding it is as easy as multiplying the angle by the radius.

### Calculus II – Arc Length

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